In this paper, we opt for a self-contained approach and ask for slightly more time regularity in our solution concept than usual in the literature on regularity theory, see [1]}, [2]}, [3]}, [4]}, [5]}.
We assume that locally the time derivative of a solution is given as a bounded linear functional on the space of test functions.
It allows us to prove a chain rule for the term involving the time derivative used to obtain the necessary estimates.
In particular, we do not assume that solutions are the limit of sufficiently regular solutions of appropriate approximate problems.
Our assumption is not restrictive and is in line with the solution concept used in [6]} and [7]}.
